Traveling Wave Solution to Filtration Model in Porous Medium

Abstract

Suspension and colloid filtration in porous rocks is encountered in problems of underground hydromechanics associated with construction. The formation of grout sediment in the pores of loose rock increases the water resistance of the soil and strengthens the foundation. A one-dimensional model of filtration in a porous medium considers the particles transport by the flow of a carrier fluid and the deposition of particles on the framework of a porous medium. The purpose of the work is to study the concentrations of suspended and settled particles of a suspension over a long time. Exact and asymptotic methods are used to obtain a solution to the model. The exact solution is presented in an implicit integral form. A set of solutions in the form of traveling waves with an arbitrary initial condition and their asymptotics are constructed. An explicit second-order asymptotic solution for a long time is obtained as an expansion in decreasing exponents. Comparison of the asymptotic solution with the traveling waves makes it possible to choose a single traveling wave corresponding to the exact solution. The closeness of the traveling wave to the exact solution of the filtration model is verified numerically. The traveling wave found determines the explicit asymptotics of the concentration of deposited particles for a long time.